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A shop owner bought a total of $64$ shirts from a wholesale market that came in two sizes, small and large. The price of a small shirt was $\text{INR} \; 50$ less than that of a large shirt. She paid a total of $\text{INR} \; 5000$ for the large shirts, and a total of $\text{INR} \; 1800$ for the small shirts. Then, the price of a large shirt and a small shirt together, in $\text{INR},$ is

1. $200$
2. $175$
3. $150$
4. $225$

Let the number of large shirts is $l$ and the number of small shirts is $s.$

$\boxed{s+l=64}$

Let the price of a small shirt be $p,$ and then the price of a large shirt be $p+50.$

Money she spent on small shirts $= \text{sp} = 1800 \; \longrightarrow (1)$

Money she spent on large shirts $= l(p+50) = 5000$

$\Rightarrow (64-s)(p+50) = 5000$

$\Rightarrow 64p + 3200 – \text{sp} – 50s = 5000$

$\Rightarrow 64p – 50s – 1800 = 1800$

$\Rightarrow 64p – 50s = 3600$

$\Rightarrow 32p – 25s = 1800$

$\Rightarrow 32p – 25\left(\frac{1800}{p}\right) = 1800$

$\Rightarrow 32p^{2} – 45000 = 1800p$

$\Rightarrow 32p^{2} – 1800p – 45000 = 0$

$\Rightarrow 4p^{2} – 225p – 5625 = 0$

$\Rightarrow p = \frac{-(-225) \pm \sqrt{(-225)^{2} – 4(4)(-5625)}}{2(4)}$

$\Rightarrow p = \frac{225 \pm \sqrt{50625+90000}}{8}$

$\Rightarrow p = \frac{225 \pm \sqrt{140625}}{8}$

$\Rightarrow p = \frac{225 \pm 375}{8}$

$\Rightarrow p = \frac{225+375}{8}, \; \frac{225-375}{8}$

$\Rightarrow p = \frac{600}{8}, \; \frac{-150}{8}$

$\Rightarrow \boxed{p=75, \; {\color{Red} {\frac{-75}{4} \;\text{(rejected)}}}}$

The price of small shirt $= p = ₹ 75.$

The price of large shirt $= p+50 = 75+50= ₹ 125.$

$\therefore$ The price of a large shirt and a small shirt together, in INR $= 75+125 =₹ 200.$

Correct Answer $:\text{A}$
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